Prediction and validation of chaotic behavior in an electrostatically ...
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PREDICTION AND VALIDATION OF CHAOTIC BEHAVIOR IN AN ELECTROSTATICALLY ACTUATED MICROELECTROMECHANICAL OSCILLATOR 1
B.E. DeMartini, H.E. Butterfield, J. Moehlis, and K.L. Turner Department of Mechanical Engineering, University of California, Santa Barbara, United States (Tel : (805) 893-7849; E-mail: [email protected])
Abstract: We investigate chaotic behavior for a microelectromechanical (MEM) oscillator, which is modeled by a version of the Mathieu equation that contains both linear and nonlinear time varying VWLIIQHVV FRHIILFLHQWV %\ XVLQJ 0HOQLNRY¶V PHWKRG ZH KDYH GHYHORSHG D FULWHULRQ IRU WKH H[LVWHQFH RI chaos in such oscillators, which depends solely on system parameters. Chaotic behavior was observed H[SHULPHQWDOO\DQGQXPHULFDOO\IRUD0(0RVFLOODWRU developed using the criterion from our analysis.
1. INTRODUCTION
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007
Keywords:FKDRV0HOQLNRY¶VPHWKRGQRQOLQHDUG\QDPLFVSDUDPHWULFUHVRQDWRUV
Many physical systems have the ability to H[KLELW FKDRWLF EHKDYLRU 3HUKDSV WKH PRVW IDPRXV H[DPSOH LV WKH /RUHQ] HTXDWLRQV >@ which have helped to understand the dynamics of cellular convection. Recently, chaos has been reported for various nonlinear MEM oscillators >@:HLQYHVWLJDWHWKHH[LVWHQFHRIFKDRVIRUD FODVV RI QRQOLQHDU SDUDPHWULFDOO\ H[FLWHG 0(0 RVFLOODWRUVWKDWKDVUHFHQWO\EHHQVWXGLHG>@DQG SURSRVHGIRUDSSOLFDWLRQVVXFKDVPDVVVHQVLQJ>@ DQGVLJQDOILOWHULQJ> @ ,QRUGHUIRUWKHVHFRPSOH[QRQOLQHDUGHYLFHVWR EH XWLOL]HG LQ UHDO ZRUOG DSSOLFDWLRQV LW LV important to understand the conditions (system parameters) that result in chaotic behavior. The ability to predict such behavior is not only useful for designing robust devices with predictable dynamics (i.e. for the applications listed above), but also for applications such as signal encryption >@WKDWH[SORLWFKDRWLFYLEUDWLRQV 2. PREDICTIVE ANALYSIS Characteristic to the class of oscillators studied herein are linear and cubic nonlinear time varying stiffness coefficients arising from the electrostatic actuation mechanism. These coefficients lead to abrupt changes in dynamic behavior (nearly instantaneous jumps from a stable quiescent state WR D VWDEOH RVFLOODWRU VWDWH ZKHQ WKH H[FLWDWLRQ IUHTXHQF\ LV QHDU Ȧ0/n (where n is an integer greater than or equal to 1), which make them attractive for applications such as those discussed LQ 6HFWLRQ $ 0(06 WKDW H[KLELWV WKLV W\SH RI
1697
behavior, shown in Figure 1, consists of noninterdigitated combdrive actuators (AC and '& VXVSHQGLQJ IOH[XUHV . DQG D VKXWWOH PDVV 0 > @
Fig.1 SEM image of the chaotic oscillator. The two sets of noninterdigitated combdrives are XVHG IRU SHULRGLF H[FLWDWLRQ $& DQG IRU '& WXQLQJ '& > @ DQG JHQHUDWH DQ HOHFWURVWDWLF force Fes [W U10[U30[3)V0 U1A[U3A[3)V(t), where r10 and r30 are electrostatic stiffness coefficients corresponding to the tuning combdrive, r1A and r3A are electrostatic stiffness FRHIILFLHQWV FRUUHVSRQGLQJ WR WKH H[FLWDWLRQ combdrive, V0 is the voltage amplitude applied to WKH '& FRPEGULYH DQG 9 W 9A FRV Ȧt)) is the signal applied to the AC combdrive (decouples SDUDPHWULF DQG KDUPRQLF H[FLWDWLRQ >@ ZLWK amplitude VA 7KH PHFKDQLFDO IOH[XUHV . generate a force, Fr [ N1[ N3[3), where k1 and k3 are stiffness coefficients. Including a dissipative force due to damping in the system, the
3EB10.P
nondimensional equation of motion is z cc D z c E z G z J 1 cos :W z 3
K 1 cos :W z 3
(1) 0
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007
where Ȧ0=(k1/m)1/2, IJ = Ȧ0t (scaled time), ȍ=Ȧ/Ȧ0 VFDOHGGULYLQJIUHTXHQF\ µ GGIJ, z = x/x0 (scaled displacement, where x0 is a characteristic length), Į =c/(Ȧ0m) (c is the damping coefficient), ȕ=1+r10V02/k1, į=x02(k3+r30V02)/k1, Ȗ=r1AVA2/k1, Ș = x02r3AVA2/k1. In this analysis we use an analytical technique, 0HOQLNRY¶V PHWKRG >@ WR GHGXFH WKH SUHVHQFH of chaos in Equation (1). In order to apply 0HOQLNRY¶VPHWKRGWRWKLVV\VWHPWKHSDUDPHWHUV are rescaled. First, parameters corresponding to time dependent and velocity terms (Į,Ȗ, and Ș) are assumed to be small (much less than 1), which is a valid assumption for MEM oscillators. In other words, they are considered perturbations to the +DPLOWRQLDQ V\VWHP ]¶¶ȕz+įz3=0. Second, the other parameters (ȕ and į) are assumed to be order one quantities. A necessary condition for this analysis is that the unperturbed Hamiltonian system must contain a double well potential, which requires (2) E 0 . Taking these assumptions into account, Melnikov DQDO\VLV >@ ZDV DSSOLHG WR GHWHUPLQH WKH following criterion for the existence of chaos § S: 3 8D E 2 G sinh ¨ ¨2 E ©
· ¸ S: 2 6GJ K 4 E : 2 . ¸ ¹
(3)
An oscillator, governed by Equation (1), whose system parameters satisfy this expression will have a chaotic invariant set, which may or may not be an attractor.
r300. Note, for the case where r30>0, į>0 will be true for all V0. To satisfy these conditions, fixed-fixed flexures (with relatively large k3) and misaligned noninterdigitated combdrives for DC WXQLQJ ZHUH FKRVHQ 6HH >@ IRU GLVFXVVLRQV on how alignment and geometry affect the sign of combdrive stiffness coefficients. Using aligned noninterdigitated combrives for AC excitation (r1A>0 and r3A@ GHYLFHVKRZQ in Figure 1), are r10= -4.7e-3 [µN/(V2µm)], r30= 1.8e-4 [µN/(V2µm3)], r1A= 1.1e-3 [µN/(V2µm)], r3A= -1.4e-4 [µN/(V2µm3)], k1= 9.6 [µN/µm], k3= 7.3 [µN/µm3], m =1.6e-9 [kg].
The electrostatic coefficients were determined using finite element software. An estimate of the quality factor was determined experimentally in a 535 mTorr vacuum environment, which is consistent throughout all experiments, to be Q = 1558. These estimated parameters were used in numerical simulations for comparison with the experimental results. A DC voltage was applied to the tuning electrodes, with no AC excitation, and was slowly increased. When V0 was large enough, the topology of the potential energy became a double well and small fabrication induced asymmetries and noise caused device to buckle to one of the new equilibrium positions. Figure 2 shows the buckling event observed in the experiment.
3. VERIFICATION OF CHAOS Both numerical and experimental investigations that verify the above criterion for the existence of chaos are presented in this section. Using Equations (2) and (3) as criteria for the parameters of the system, an oscillator was designed to exhibit chaotic behavior. In order to satisfy ȕ@( /RUHQ] ³'HWHUPLQLVWLF QRQSHULRGLF IORZ´ J. Atm. Science, Vol. 20, pp. 130-141, 1963. >@0%DVVR/*LDUUH0'DKOHKDQG,0H]LF ³1XPHULFDO DQDO\VLV RI FRPSOH[ G\QDPLFV LQ DWRPLF IRUFH PLFURVFRSHV´ ,((( Intnl. Conf. on Control. Apps., Trieste, Italy, 1-4 September, 1998. >@@$ /XR DQG):DQJ³1RQOLQHDU G\QDPLFVRI a micro-electro-mechanical system with timeYDU\LQJ FDSDFLWRUV´ - 9LE DQG $FRXVW YRO 126, pp. 77-83, 2004. [6] K. Turner, S. Miller, P. Harwell, N. 0DF'RQDOG66WURJDW]DQG6$GDPV³)LYH parametric resonances in a PLFURHOHFWURPHFKDQLFDO V\VWHP´ 1DWXUH 9RO 396, pp. 149-152, 1998. [7] J. Rhoads, S. Shaw, K. Turner, J. Moehlis, B. 'H0DUWLQL DQG : =KDQJ ³*HQHUDOL]HG parametric resonance in electrostaticallyDFWXDWHG PLFURHOHFWURPHFKDQLFDO RVFLOODWRUV´ J. Sound and Vib., Vol. 296, pp. 797-829, 2006. >@:=KDQJ5%DVNDUDQDQG.7XUQHU³(IIHFW of nonlinearity on auto-parameterically DPSOLILHGUHVRQDQW0(06PDVVVHQVRU´6HQV and Act. A-Phys., Vol. 103 No. 1-2, pp. 139150, 2002. [9] J. Rhoads, S. Shaw, K. Turner, and R. %DVNDUDQ³7XQDEOH0(06ILOWHUVWKDWH[SORLW SDUDPHWULF UHVRQDQFH´ - 9LE DQG $FRXVW Vol. 127, No. 5, pp. 423-430, 2004. [10] S.G. Adams, F.M. Bertsch, K.A. Shaw, and 1& 0DF'RQDOG ³,QGHSHQGHQW WXQLQJ RI OLQHDU DQG QRQOLQHDU VWLIIQHVV FRHIILFLHQWV´ - MEMS, Vol. 7, No. 2, pp. 172-180, 1998. >@%'H0DUWLQL-5KRDGV.7XUQHU66KDZ DQG - 0RHKOLV ³/LQHDU DQG QRQOLQHDU WXQLQJ RISDUDPHWULFDOO\H[FLWHG0(0RVFLOODWRUV´,Q Press, J. MEMS, 2007. [12] J. Guckenheimer and P. Holmes, Nonlinear 2VFLOODWRUV '\QDPLFDO 6\VWHPV DQG Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. >@ : =KDQJ : =KDQJ . 7XUQHU DQG 3 +DUWZHOO ³6&5($0¶ $ VLQJOH PDVN SURFHVV IRU KLJK4 VLQJOH FU\VWDO VLOLFRQ 0(06´$60( ,QWQO 0HFK (QJ &RQJU RQ Sens., Vienna, Austria, 24-27 Oct. 2004.
PREDICTION AND VALIDATION OF CHAOTIC BEHAVIOR IN AN ELECTROSTATICALLY ACTUATED MICROELECTROMECHANICAL OSCILLATOR 1
B.E. DeMartini, H.E. Butterfield, J. Moehlis, and K.L. Turner Department of Mechanical Engineering, University of California, Santa Barbara, United States (Tel : (805) 893-7849; E-mail: [email protected])
Abstract: We investigate chaotic behavior for a microelectromechanical (MEM) oscillator, which is modeled by a version of the Mathieu equation that contains both linear and nonlinear time varying VWLIIQHVV FRHIILFLHQWV %\ XVLQJ 0HOQLNRY¶V PHWKRG ZH KDYH GHYHORSHG D FULWHULRQ IRU WKH H[LVWHQFH RI chaos in such oscillators, which depends solely on system parameters. Chaotic behavior was observed H[SHULPHQWDOO\DQGQXPHULFDOO\IRUD0(0RVFLOODWRU developed using the criterion from our analysis.
1. INTRODUCTION
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007
Keywords:FKDRV0HOQLNRY¶VPHWKRGQRQOLQHDUG\QDPLFVSDUDPHWULFUHVRQDWRUV
Many physical systems have the ability to H[KLELW FKDRWLF EHKDYLRU 3HUKDSV WKH PRVW IDPRXV H[DPSOH LV WKH /RUHQ] HTXDWLRQV >@ which have helped to understand the dynamics of cellular convection. Recently, chaos has been reported for various nonlinear MEM oscillators >@:HLQYHVWLJDWHWKHH[LVWHQFHRIFKDRVIRUD FODVV RI QRQOLQHDU SDUDPHWULFDOO\ H[FLWHG 0(0 RVFLOODWRUVWKDWKDVUHFHQWO\EHHQVWXGLHG>@DQG SURSRVHGIRUDSSOLFDWLRQVVXFKDVPDVVVHQVLQJ>@ DQGVLJQDOILOWHULQJ> @ ,QRUGHUIRUWKHVHFRPSOH[QRQOLQHDUGHYLFHVWR EH XWLOL]HG LQ UHDO ZRUOG DSSOLFDWLRQV LW LV important to understand the conditions (system parameters) that result in chaotic behavior. The ability to predict such behavior is not only useful for designing robust devices with predictable dynamics (i.e. for the applications listed above), but also for applications such as signal encryption >@WKDWH[SORLWFKDRWLFYLEUDWLRQV 2. PREDICTIVE ANALYSIS Characteristic to the class of oscillators studied herein are linear and cubic nonlinear time varying stiffness coefficients arising from the electrostatic actuation mechanism. These coefficients lead to abrupt changes in dynamic behavior (nearly instantaneous jumps from a stable quiescent state WR D VWDEOH RVFLOODWRU VWDWH ZKHQ WKH H[FLWDWLRQ IUHTXHQF\ LV QHDU Ȧ0/n (where n is an integer greater than or equal to 1), which make them attractive for applications such as those discussed LQ 6HFWLRQ $ 0(06 WKDW H[KLELWV WKLV W\SH RI
1697
behavior, shown in Figure 1, consists of noninterdigitated combdrive actuators (AC and '& VXVSHQGLQJ IOH[XUHV . DQG D VKXWWOH PDVV 0 > @
Fig.1 SEM image of the chaotic oscillator. The two sets of noninterdigitated combdrives are XVHG IRU SHULRGLF H[FLWDWLRQ $& DQG IRU '& WXQLQJ '& > @ DQG JHQHUDWH DQ HOHFWURVWDWLF force Fes [W U10[U30[3)V0 U1A[U3A[3)V(t), where r10 and r30 are electrostatic stiffness coefficients corresponding to the tuning combdrive, r1A and r3A are electrostatic stiffness FRHIILFLHQWV FRUUHVSRQGLQJ WR WKH H[FLWDWLRQ combdrive, V0 is the voltage amplitude applied to WKH '& FRPEGULYH DQG 9 W 9A FRV Ȧt)) is the signal applied to the AC combdrive (decouples SDUDPHWULF DQG KDUPRQLF H[FLWDWLRQ >@ ZLWK amplitude VA 7KH PHFKDQLFDO IOH[XUHV . generate a force, Fr [ N1[ N3[3), where k1 and k3 are stiffness coefficients. Including a dissipative force due to damping in the system, the
3EB10.P
nondimensional equation of motion is z cc D z c E z G z J 1 cos :W z 3
K 1 cos :W z 3
(1) 0
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007
where Ȧ0=(k1/m)1/2, IJ = Ȧ0t (scaled time), ȍ=Ȧ/Ȧ0 VFDOHGGULYLQJIUHTXHQF\ µ GGIJ, z = x/x0 (scaled displacement, where x0 is a characteristic length), Į =c/(Ȧ0m) (c is the damping coefficient), ȕ=1+r10V02/k1, į=x02(k3+r30V02)/k1, Ȗ=r1AVA2/k1, Ș = x02r3AVA2/k1. In this analysis we use an analytical technique, 0HOQLNRY¶V PHWKRG >@ WR GHGXFH WKH SUHVHQFH of chaos in Equation (1). In order to apply 0HOQLNRY¶VPHWKRGWRWKLVV\VWHPWKHSDUDPHWHUV are rescaled. First, parameters corresponding to time dependent and velocity terms (Į,Ȗ, and Ș) are assumed to be small (much less than 1), which is a valid assumption for MEM oscillators. In other words, they are considered perturbations to the +DPLOWRQLDQ V\VWHP ]¶¶ȕz+įz3=0. Second, the other parameters (ȕ and į) are assumed to be order one quantities. A necessary condition for this analysis is that the unperturbed Hamiltonian system must contain a double well potential, which requires (2) E 0 . Taking these assumptions into account, Melnikov DQDO\VLV >@ ZDV DSSOLHG WR GHWHUPLQH WKH following criterion for the existence of chaos § S: 3 8D E 2 G sinh ¨ ¨2 E ©
· ¸ S: 2 6GJ K 4 E : 2 . ¸ ¹
(3)
An oscillator, governed by Equation (1), whose system parameters satisfy this expression will have a chaotic invariant set, which may or may not be an attractor.
r300. Note, for the case where r30>0, į>0 will be true for all V0. To satisfy these conditions, fixed-fixed flexures (with relatively large k3) and misaligned noninterdigitated combdrives for DC WXQLQJ ZHUH FKRVHQ 6HH >@ IRU GLVFXVVLRQV on how alignment and geometry affect the sign of combdrive stiffness coefficients. Using aligned noninterdigitated combrives for AC excitation (r1A>0 and r3A@ GHYLFHVKRZQ in Figure 1), are r10= -4.7e-3 [µN/(V2µm)], r30= 1.8e-4 [µN/(V2µm3)], r1A= 1.1e-3 [µN/(V2µm)], r3A= -1.4e-4 [µN/(V2µm3)], k1= 9.6 [µN/µm], k3= 7.3 [µN/µm3], m =1.6e-9 [kg].
The electrostatic coefficients were determined using finite element software. An estimate of the quality factor was determined experimentally in a 535 mTorr vacuum environment, which is consistent throughout all experiments, to be Q = 1558. These estimated parameters were used in numerical simulations for comparison with the experimental results. A DC voltage was applied to the tuning electrodes, with no AC excitation, and was slowly increased. When V0 was large enough, the topology of the potential energy became a double well and small fabrication induced asymmetries and noise caused device to buckle to one of the new equilibrium positions. Figure 2 shows the buckling event observed in the experiment.
3. VERIFICATION OF CHAOS Both numerical and experimental investigations that verify the above criterion for the existence of chaos are presented in this section. Using Equations (2) and (3) as criteria for the parameters of the system, an oscillator was designed to exhibit chaotic behavior. In order to satisfy ȕ@( /RUHQ] ³'HWHUPLQLVWLF QRQSHULRGLF IORZ´ J. Atm. Science, Vol. 20, pp. 130-141, 1963. >@0%DVVR/*LDUUH0'DKOHKDQG,0H]LF ³1XPHULFDO DQDO\VLV RI FRPSOH[ G\QDPLFV LQ DWRPLF IRUFH PLFURVFRSHV´ ,((( Intnl. Conf. on Control. Apps., Trieste, Italy, 1-4 September, 1998. >@@$ /XR DQG):DQJ³1RQOLQHDU G\QDPLFVRI a micro-electro-mechanical system with timeYDU\LQJ FDSDFLWRUV´ - 9LE DQG $FRXVW YRO 126, pp. 77-83, 2004. [6] K. Turner, S. Miller, P. Harwell, N. 0DF'RQDOG66WURJDW]DQG6$GDPV³)LYH parametric resonances in a PLFURHOHFWURPHFKDQLFDO V\VWHP´ 1DWXUH 9RO 396, pp. 149-152, 1998. [7] J. Rhoads, S. Shaw, K. Turner, J. Moehlis, B. 'H0DUWLQL DQG : =KDQJ ³*HQHUDOL]HG parametric resonance in electrostaticallyDFWXDWHG PLFURHOHFWURPHFKDQLFDO RVFLOODWRUV´ J. Sound and Vib., Vol. 296, pp. 797-829, 2006. >@:=KDQJ5%DVNDUDQDQG.7XUQHU³(IIHFW of nonlinearity on auto-parameterically DPSOLILHGUHVRQDQW0(06PDVVVHQVRU´6HQV and Act. A-Phys., Vol. 103 No. 1-2, pp. 139150, 2002. [9] J. Rhoads, S. Shaw, K. Turner, and R. %DVNDUDQ³7XQDEOH0(06ILOWHUVWKDWH[SORLW SDUDPHWULF UHVRQDQFH´ - 9LE DQG $FRXVW Vol. 127, No. 5, pp. 423-430, 2004. [10] S.G. Adams, F.M. Bertsch, K.A. Shaw, and 1& 0DF'RQDOG ³,QGHSHQGHQW WXQLQJ RI OLQHDU DQG QRQOLQHDU VWLIIQHVV FRHIILFLHQWV´ - MEMS, Vol. 7, No. 2, pp. 172-180, 1998. >@%'H0DUWLQL-5KRDGV.7XUQHU66KDZ DQG - 0RHKOLV ³/LQHDU DQG QRQOLQHDU WXQLQJ RISDUDPHWULFDOO\H[FLWHG0(0RVFLOODWRUV´,Q Press, J. MEMS, 2007. [12] J. Guckenheimer and P. Holmes, Nonlinear 2VFLOODWRUV '\QDPLFDO 6\VWHPV DQG Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. >@ : =KDQJ : =KDQJ . 7XUQHU DQG 3 +DUWZHOO ³6&5($0¶ $ VLQJOH PDVN SURFHVV IRU KLJK4 VLQJOH FU\VWDO VLOLFRQ 0(06´$60( ,QWQO 0HFK (QJ &RQJU RQ Sens., Vienna, Austria, 24-27 Oct. 2004.
